the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Jul 2026
Mathematical Modelling of Sediment Thickness and Bedrock Topography Using the Biharmonic Equation and Constrained Sequential Gaussian Simulation: Bi-cSGS-Surface-v1.0
Abstract. In this study, we present a two-stage surface reconstruction framework in which the target surface is modeled as a spatial random function composed by a trend function and stochastic residual component. The trend is recovered by solving the discrete form of a biharmonic equation, ensuring a smooth representation of the large scale structure, while the residual is simulated using constrained Sequential Gaussian Simulation (cSGS) to reproduce small scale spatial variability. Bounds are imposed during simulation to maintain consistency with geomorphological constraints.
The approach is implemented using measurement data from wells, along with Digital Elevation Models and digital quaternary maps as input. Two strategies are evaluated: cSGS applied directly to the observed variables, and cSGS applied to the residuals after subtracting the biharmonic trend. Variogram models are fitted to experimental variograms obtained from the data, and model performance is assessed using an independent dataset. Cross validation on an independent dataset shows that explicitly incorporating the trend component prior to stochastic simulation improves predictive accuracy, yielding lower mean absolute error and stochastic simulation results better centered around observations.
The methodology demonstrates the importance of modeling large-scale trends before cSGS and provides a flexible, physically consistent framework for estimating sediment thickness and bedrock topography under sparse data and uncertainty.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-3214', Bernd Simeon, 31 Aug 2026
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AC2: 'Reply on RC1', Maregnesh Mechal Wolde, 16 Sep 2026
We thank the referee for this important comment. We agree that error bounds would be desirable. However,
the fundamental question is how the error should be defined and in which norm it should be measured.
The appropriate definition of the error depends on the purpose of the application.
For example, the bedrock topography may be important for construction applications,
where a pointwise measure of accuracy is relevant. In contrast, agricultural applications
may require estimates of the volume of water that can be stored in a sediment layer, for which
an averaged error measure may be more appropriate. The distinction lies in whether one is interested
in pointwise accuracy or in integral or averaged quantities.Furthermore, the use of a discrete biharmonic approach is not intended to approximate the solution
of the continuous biharmonic equation. Rather, the objective is to construct a smooth large-scale surface using
a finite number of model parameters. Consequently, the mesh size is not a critical parameter in our
simulations, the amount and spatial distribution of the data are much more important for the present application.The key relationship is between the number of data points and the number of model degrees of
freedom. This ratio, rather than the mesh resolution itself, governs the quality and reliability of
the reconstructed surface.Citation: https://doi.org/10.5194/egusphere-2026-3214-AC2
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AC2: 'Reply on RC1', Maregnesh Mechal Wolde, 16 Sep 2026
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RC2: 'Comment on egusphere-2026-3214', Anonymous Referee #2, 03 Sep 2026
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AC1: 'Reply on RC2', Maregnesh Mechal Wolde, 13 Sep 2026
Dear Referee,
Thank you for your detailed comments and for the opportunity to clarify the methodology and the role of conditional constraints in our approach.
We have prepared a detailed response addressing the points raised in the comment. The response is provided in the attached PDF.
We will incorporate the relevant clarifications into the manuscript following the public discussion phase.Best regards,
Maregnesh Mechal Wolde
on behalf of the authors
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AC1: 'Reply on RC2', Maregnesh Mechal Wolde, 13 Sep 2026
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EC1: 'Comment on egusphere-2026-3214', Thomas Poulet, 23 Sep 2026
Dear authors,
Having considered both reports carefully, I believe that the concern raised by Reviewer #2 merits particular attention. At the same time, I consider that the manuscript still has the potential to make a valuable contribution if the authors can provide a clear and convincing response to this issue, supported where necessary by additional analysis, validation, discussion, or revisions to the methodology.
I therefore encourage the authors to submit a revised manuscript. In preparing the revision, please pay especially close attention to Reviewer #2's main comment and ensure that the revised manuscript and response letter address this point comprehensively and transparently. The success of any resubmission will depend largely on whether this concern can be satisfactorily resolved.
Best regards,
Thomas Poulet
Citation: https://doi.org/10.5194/egusphere-2026-3214-EC1
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This work combines the fields of mathematical modelling, numerical analysis and stochastic data analysis with an interesting application in geoscience. More specifically, it provides an approach to construct a smooth approximation to the distribution of the sediment thickness and bedrock topography by means of a model from partial differential equations, the biharmonic or plate equation. This PDE model is solved by state-of-the-art virtual element methods. The stochastic part combines the smooth global model with sequential Gaussian simulation to extend the data from local wells as stochastic variations. Extensive numerical studies and validations with data from Norwegian geological explorations confirm the results.
In my point of view, this work contains several important contributions. Using a plate model for constructing a smooth approximation of the thickness distribution is an innovative approach, and it requires quite some detailed work on appropriate boundary conditions and on connecting the source term to given data points. Moreover, the modification of sequential Gaussian simulation to include constraints and its combination with the plate model constitute a remarkable progress that has the potential to reach far beyond the field of geoscience. It is quite impressive how well the methodology works for the Norwegian data.
Issues and open questions: The submission is carefully written and makes adequate references to related work. But for me as a mathematician, it would be desirable to have theoretical error bounds for the overall approach that combines the plate or biharmonic model with the sequential Gaussian simulation. Such bounds would further underline the impact of the results. If a theoretical study is too hard to achieve, numerical studies with different mesh sizes for the biharmonic equation would also be beneficial.